How to get the UV corresponding to a vertex via the Python API?
As madlaina says, A vertex doesn't have a 1:1 corraspondance with UV coordinates.

However using BMesh you can loop over a vertices connected face corners (loops), and find the UV's used by a vertex.

Heres a script that prints all vertices UV's in editmode.

Both the first-uv found (which is OK if you assume all UV's attached to a vertex will be the same).
And the mean average.

import bpy
import bmesh
from mathutils import Vector

obj = bpy.context.edit_object
me = obj.data
bm = bmesh.from_edit_mesh(me)


def uv_from_vert_first(uv_layer, v):
    for l in v.link_loops:
        uv_data = l[uv_layer]
        return uv_data.uv
    return None


def uv_from_vert_average(uv_layer, v):
    uv_average = Vector((0.0, 0.0))
    total = 0.0
    for loop in v.link_loops:
        uv_average += loop[uv_layer].uv
        total += 1.0

    if total != 0.0:
        return uv_average * (1.0 / total)
    else:
        return None

# Example using the functions above
uv_layer = bm.loops.layers.uv.active

for v in bm.verts:
    uv_first = uv_from_vert_first(uv_layer, v)
    uv_average = uv_from_vert_average(uv_layer, v)
    print("Vertex: %r, uv_first=%r, uv_average=%r" % (v, uv_first, uv_average))